Theorem dfpss3 3030

Description: Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
dfpss3	|- ( A C. B <-> ( A C_ B /\ -. B C_ A ) )

Proof of Theorem dfpss3

Step	Hyp	Ref	Expression
1		dfpss2 3029	. 2 |- ( A C. B <-> ( A C_ B /\ -. A = B ) )
2		eqss 2960	. . . . 5 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
3	2	baib 828	. . . 4 |- ( A C_ B -> ( A = B <-> B C_ A ) )
4	3	notbid 592	. . 3 |- ( A C_ B -> ( -. A = B <-> -. B C_ A ) )
5	4	pm5.32i 427	. 2 |- ( ( A C_ B /\ -. A = B ) <-> ( A C_ B /\ -. B C_ A ) )
6	1, 5	bitri 173	1 |- ( A C. B <-> ( A C_ B /\ -. B C_ A ) )

Syntax hints:   -. wn 3   /\ wa 97   <-> wb 98   = wceq 1243   C_ wss 2917   C. wpss 2918

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ne 2206  df-in 2924  df-ss 2931  df-pss 2933

This theorem is referenced by:  pssirr  3044  pssn2lp  3045  ssnpss  3047  nsspssun  3170  npss0  3266